Keywords: wave propagation, layered structure, optimization, impulse loading.

In this ongoing research, we are developing new material architectures for mitigating the effects of impulsive loadings using functionally graded materials (FGM). Stress attenuation is obtained by tuning material properties and length of each layer, along with the total length of a multi-layer structure utilizing the geometric dispersion [1] found in layered structures.

A number of researchers have looked at wave propagation in layered structures. For example, Robnik [2] proposed an analytic approximation method using the Wentzel-Kramers-Brillouin (WKB) approximation, which may be applied to discrete and continuous media. Tenenbaum and Zindeluk [3] proposed an approach to the direct scattering problem in one-dimensional inhomogeneous bodies. Nygren et al. [4] presented an optimization analysis of elastic junctions with regard to transmission of wave energy of an incident extensional wave at a non-uniform elastic junction between two uniform and collinear elastic bars. Gusev [5,6] studied the optimal synthesis of multilayer structures by implementing the ultimate performance under the action of elastic waves.

In contrast to the body of literature on time harmonic wave propagation, there is much less published work on the optimization of layered structures subjected to transient loading. Anfinsen [7] studied the problem of maximizing or minimizing the amplitude of stress waves propagating through a one-dimensional elastic layered structure using difference equations, which were solved using z-transform methods. Some other works on transient optimal design of multi-layer structures can been found in References [8-10].

Perhaps the closest work that has been done by other researchers is that by Velo and Gazonas [11]. They investigated the optimal design of a two-layered elastic strip subjected to transient loading by assuming a Goupillaud-type layered medium. The chosen design parameter was the impedance ratio between the two layers. However, there are additional control parameters (i.e., length of each layer) that need to be determined to obtain the final optimal design even after the optimal impedance ratio is found.

In this paper, an optimal design procedure is proposed to obtain the optimal length of each layer. First, the stress transfer function is obtained for a multi-layer structure with one fixed end boundary condition based on a transfer matrix method. Given this stress transfer function, stress output at the fixed end could be computed under any arbitrary transient load. Then, analytical solutions of stress at the fixed surface is obtained for both two-layered Goupillaud-type media and a large second layer structure using an inverse Laplace transform method. Optimal criteria is then proposed and the optimal structure is determined for this two-layered structure, based upon an analysis of the stress at the fixed end. The analysis shows that the optimal solution depends on the applied load, however, the optimal procedure works for any arbitrary transient load. By applying the load and following the optimal procedure proposed in this paper, the optimal structure can be achieved. Finite element simulations are also carried out using the commercial code ABAQUS [12] for comparison with the analytical results.

[1]

J.C. Peck, "Pulse attenuation in composites", Proc., 17th Sagamore Army Materials Research Conference, Raquette Lake, New York, 155-184, 1970.

[2]

M. Robnik, "Matrix treatment of wave propagation in stratified media", Journal of Physics A: Mathematical and General, 12(1), 151-158, 1979. doi:10.1088/0305-4470/12/1/028

[3]

R.A. Tenenbaum and M. Zindeluk, "An exact solution for the one-dimensional elastic wave equation in layered media", Journal of Acoustical Society of America, 92(6), 3364-3370, 1992. doi:10.1121/1.404186

[4]

T. Nygren, L. E. Andersson and B. Lundberg, "Optimization of elastic junctions with regard to transmission of wave energy", Wave Motion, 29, 223-244, 1999. doi:10.1016/S0165-2125(98)00041-9

[5]

E.L. Gusev, "Internal structural symmetry of optimal layered structures", Acoustical Physics, 47(1), 56-61, 2001. doi:10.1134/1.1340078

[6]

E.L. Gusev, "Optimal synthesis of inhomogeneous layered structures", Acoustical Physics, 48(3), 56-61, 2002. doi:10.1134/1.1478111

[7]

L.E. Anfinsen, "Optimum design of layered elastic stress wave attenuators", Journal of Applied Mechanics, 34, 751-755, 1967.

[8]

H. Kongstanty and F. Santo, "Optimal design of minimally reflective coatings", Wave Motion, 21, 291-309, 1995. doi:10.1016/0165-2125(95)00004-3

[9]

A. Karlsson, "Wave propagators for transient waves in one-dimensional media", Wave Motion, 24, 85-99, 1996. doi:10.1016/0165-2125(96)00008-X

[10]

R. Laverty and G. Gazonas, "Optimal design of multi-layered stress wave attenuators", Proc., 13th Annual ARL/USMA Technical Symposium, West Point, New York, 2005.

[11]

A.P. Velo and G.A. Gazonas, "Optimal design of a two-layered elastic strip subjected to transient loading", International Journal of Solids and Structures, 40, 6417-6428, 2003. doi:10.1016/S0020-7683(03)00438-4

[12]

Hibbitt, Karlsson & Sorensen, Inc, "ABAQUS/STANDARD User's Manual", Version 6.3, 2003.

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